Vol. 88, No. 2, 1980

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Nonexistence of F-minimizing embedded disks

Jean Ellen Taylor

Vol. 88 (1980), No. 2, 279–283
Abstract

There has been considerable interest recently in the question of when, given a smooth simple closed extreme curve in R3 as boundary, there exists an embedding of a disk having that boundary which is minimal or minimizing in some appropriate subclass of Lipschitz mappings of the disk. Almgren and Simon [2] showed that an area minimizing embedding of a disk exists in the class of all Lipschitz embeddings of disks. (They also showed that there exists an area minimizing embedding of a disk with k handles in the class of all such embeddings in case there exists some mapping of the disk with k handles whose area is less than that of any mapping with k 1 handles.) Tomi and Tromba [6] showed that there exists a minimal (not necessarily minimizing) embedding of a disk in the class of all Lipschitz mappings of the disk. Meeks and Yau [3] have shown that there exists an area minimizing embedding of the disk in the class of all Lipschitz mappings of the disk.

This paper shows that if one minimizes the integral of an essentially oriented integrand, it is possible for an immersion of the disk to have less integral than any embedding; such integrands arbitrarily closely approximate area.

Mathematical Subject Classification
Primary: 49F25, 49F25
Milestones
Received: 24 July 1979
Published: 1 June 1980
Authors
Jean Ellen Taylor